Preheat the oven to 350 degrees. Prepare a 9” round springform pan by spraying it with pan spray and lining the bottom with parchment paper.
Begin by assembling the streusel, and then make the apricot butter. You’ll finish by preparing the cake batter, and then assemble the coffee cake from there.
Make the streusel. Assemble all the dry ingredients in a small bowl and whisk to combine. Drizzle in the olive oil and vanilla extract. Stir the mixture with a fork until it forms a crumbly texture - if it still appears like there are dry patches of flour, add a little more olive oil a teaspoon at a time until it comes together. Spread the streusel in a single layer on a sheet tray and pop it in the freezer while you assemble the rest of your coffee cake elements.
Make the apricot butter. Simmer the apricots, water, sugar, and lemon juice together in a small non reactive pot until the sugar has dissolved and the apricots have become very tender, about three minutes. Puree the mixture until very smooth - you may need to add a little water if the mixture is quite thick. Let cool to room temperature.
Make the cake batter. Combine the flour, pistachios, sugar, baking powder, baking soda, and salt in the bowl of a stand mixer and whisk to combine. Add the room-temperature butter to the dry ingredients and mix with a paddle attachment until the mixture looks coarse and crumbly, like sand.
Combine the sour cream, olive oil, egg, egg yolk, and vanilla in a separate bowl and whisk to combine. Add half the wet ingredients to the dry ingredients and mix at medium-low speed until the wet ingredients are fully incorporated and the mixture takes on a lightened appearance, about two minutes. Scrape down the sides of the bowl. Add the remaining sour cream mixture to the bowl and mix for an additional three minutes at medium speed.
Spread the cake batter into the prepared pan. Sprinkle with half of the streusel. Gently spoon the apricot mixture onto the surface of the cake batter - or if you have a piping bag, it can be helpful to use one to pipe the apricot butter evenly over the surface of the cake batter. Use a spoon or an offset spatula to gently smooth the apricot butter layer over the surface of the coffee cake batter. Finish with the remaining pistachio and olive oil streusel.
Bake for 40-42 minutes or until the cake has taken on a burnished golden brown color around the border. The thick layer of apricot butter can make it a little tricky to tell if the cake is done, as a knife poked in the center can still come out moist. I like to look at visual indicators here - the center of your coffee cake shouldn’t wobble when gently shaken, and the border of the cake will have separated from the sides of the pan. You’ll want to err on the side of slightly underdone rather than over.
Allow to cool completely before slicing, or let the cake rest, loosely tented in foil, overnight to enjoy the next day.